Hodge Theory and Symplectic Boundary Conditions

TL;DR

This paper develops Hodge theory for symplectic manifolds with boundary by introducing new boundary conditions, enabling the study of symplectic cohomologies, harmonic forms, and boundary value problems in this setting.

Contribution

It introduces novel boundary conditions for symplectic Laplacians, establishing Hodge decompositions and isomorphisms for symplectic cohomologies on manifolds with boundary.

Findings

Established Hodge decomposition for symplectic Laplacians with boundary conditions

Proved isomorphisms between symplectic cohomologies and harmonic fields

Applied results to solve boundary value problems for differential forms

Abstract

We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various form decomposition and also isomorphisms between the symplectic cohomologies and the spaces of harmonic fields. These novel boundary conditions can be applied in certain cases to study relative symplectic cohomologies and Lefschetz maps between relative de Rham cohomologies. As an application, our results are used to solve boundary value problems of differential forms.